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Differentials Differentials are a powerful mathematical tool They require, however, precise introduction In particular we have to distinguish between exact inexact differentials Remember some important mathematical background Multi-Dimensional Spaces Point ( in D-dimensional space ) Line: parametric representationD functions depending on a single parameter

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for Example in D=2 from classical mechanics x2x2 x1x1 where t 0 =0 and t f =2v y /g 0

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Scalar field: a single function of D coordinates For example: the electrostatic potential of a charge or the gravitational potential of the mass M (earth for instance) r

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Vector field: specified by the D components of a vector. Each component is a function of D coordinates Graphical example in D=3 Well-known vector fields in D=3 Force F(r) in a gravitational field Electric field: E(r) Magnetic field: B(r) x y z Each point in space 3 component entity

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Line integral: scalar product If the line has the parameter representation:i=1,2,…,D for The line integral can be evaluated like an ordinary 1-dimensional definite Integral

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Let’s explore an example: Consider the electric field created by a changing magnetic field x y z where t x y x y x y 0 f R Line of integration

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Parameter representation of the line: Counter clockwise walk along the semicircle of radius R yx 1 Note: Result is independent of the parameterization

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Let’s also calculate the integral around the full circle: x y R Line of integration Parameter representation of the line: Faraday’s law of electrodynamics Have a closer look to Differential form or

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Meaning of an equation that relates one differential form to another Equation valid for all lines Must be true for all sets of coordinate differentials Example: Particular set of differentials Relationships valid for vector fields are also valid for differentials

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A differential form is an exact differential. Exact and Inexact Differentials if for all i and j it is true that An equivalent condition reads: also written as Let’s do these Exactness tests in the case of our example Is the differential form x y z t exact

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Check of the cross-derivatives but Not exact Alternatively we can also show: = - + = = 0

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T, V are the coordinates of the space Example from thermodynamics Exactness of 1 Transfer of notation: Functions corresponding to the vector components: Check of the cross-derivatives 2 = exact

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Differential of a function Scalar field: a single function of D coordinates or in compact notation where Differentials of functions are exact Proof: Or alternatively: Line integral of a differential of a function x1x1 x2x2 Independent of the path between and

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We are familiar with this property from varies branches of physics: Conservative forces: Remember: A force which is given by the negative gradient of a scalar potential is known to be conservative Example: Gravitational force derived from hh Pot. energy depends on h, not how to get there.

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Exact differential theorem 1 The following 4 statements imply each other dA is the differential of a function 2 dA is exact 3 for all closed contours 4 Independent of the line connecting and x1x1 x2x2

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How to find the function underlying an exact differential Consider:Since dA exact Aim:Find A(x,y) by integration Unknown function depending on y only constant Unknown function depending on x only constant C o m p a r i s o n A(x,y) Apart from one const.

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Example: where a,b and c are constants First we check exactness Comparison Check:

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Quantities of infinitesimal short sub-processes Inexact Differentials of Thermodynamics Equilibrium processes can be represented by lines in state spaceWe know: Consider infinitesimal short sub-process Values of W,Q and are Since U is a state function we can expressU=U(T,V) dU differential form of a functiondU exact However: inexact With first law for all lines L

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How can we see that inexact Compare with the general differential form for coordinates P and V and = inexact Example: Line dependence of W and line independence of U P0P0 PfPf V0V0 VfVf Work: isothermal

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Coordinates on common isotherm = Internal energy: 1 Isothermal process from Ideal gas U=U(T) 2 P V Across constant volume and constant pressure path P0P0 PfPf V0V0 VfVf 1 2

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Sinceinexact How can we see that is inexact Consider U=U(P,V)where P and V are the coordinates with T 0 = T f -R+R Alternatively inspection of exact inexact

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Coordinate transformations Example: Changing coordinates of state space from (P,V)(T,P) V=V(T,P) If U=U(T,P) With +

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Let’s collect terms of common differentials Remember: EnthalpyH=U+PVwith Similar for changing coordinates of state space from (P,V)(T,V)

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Heat capacities expressed in terms of differentials From and P=const. V=const. Note:and are alternate notation for the components and ( of the above vector fields which correspond to the differential forms ) Do not confuse with partial derivatives, since there is no function Q(T,P) whose differential is. is inexact

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